President McKenna: What's that?
William Stryker: A jet.
President McKenna: What kind of jet?
William Stryker: We don't know...but it comes out of the basketball court.
When the soldiers first enter the mansion, and the little kid who doesn't sleep goes to investigate - listen to the commentary on the nature documentary he's watching - it fits with events quite nicely for a while, along the lines of "With the mother away, the children are left defenseless..." It goes a bit awry once it starts with "the centipede is deadly", but you get the idea. See more...
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Continuity: In the end, when the President is visited by the X-Men, he receives a blue binder, which is laid in front of him on the table. When they have left, you see a shot including the President's desk, and you can see that the only thing he has on the table are some sheets of paper, stapled in the upper left corner, opened up. Then the shot changes to a close-up of the President, and then back again, and you see the blue binder in front of him, closed, and the papers have disappeared.
Continuity: When Wolverine attacks the SWAT member in the Mutant Academy kitchen while Iceman takes cover, there's a spot the difference competition between the start and end of the gunfight. After a second or two there are bullet holes in the back wall, the Dr. Pepper's been knocked over, the basket's gone from the corner of the table, and the chair at the right is maybe a foot from the corner. When we cut back, the bullet holes in the wall have changed position, the Dr. Pepper's righted itself, the basket's back on the corner, and the chair's now much closer to the edge. All this while Iceman's hiding and Wolverine and the gunman are fighting each other.
Revealing: In the scene where Storm and Nightcrawler rush to rescue the children from the holding cell, you see a shot of the cell where the children stand up and say "Listen..." before Nightcrawler teleports inside. If you look carefully, you will see Nightcrawler's silhouette in the shadow against the wall.
Deliberate "mistake": Almost trivia, but not quite. When Mystique's looking through the guard files to find a suitable candidate, lots of photos flash up very quickly before she finds the guy she uses. Of note is that Bryan Singer pops up in guard uniform (with his own name), and so does someone called "Donner, L". This is most likely referring to Lauren Shuler Donner, one of the producers, and sex is specified as "F", but the photo is of a stubble-coated man.
Other: When the X-Mansion is being attacked, Wolverine jumps off the stairs onto two soldiers and throws them backwards with his claws. The soldiers are level with his wrists as they begin flying back. If they had been thrown in such a way, they would have followed the same path through the air as Wolverine's claws.
Continuity: When Wolverine, Rogue, Bobby and John are in Bobby's house and are talking to his parents, his dad says to Wolverine "What are you a professor of?" Just after this the camera cuts to an overhead shot, there is a solid orange pillow at Bobby's brother's side, but in all following shots the solid orange pillow has vanished and is replaced by two striped pillows.
Continuity: When Mystique is looking around in Stryker's computer, she notices a folder marked "Cerebro", next to the "Delete" icon on the screen. However, when she opens the folder, we cut to a wide shot and the icons around the delete icon have renamed themselves, plus the cerebro folder itself has disappeared.